Entropy and dimension of disintegrations of stationary measures

We extend a result of Ledrappier, Hochman, and Solomyak on exact dimensionality of stationary measures for SL 2 ( R ) \text {SL}_2(\mathbb {R}) to disintegrations of stationary measures for GL ⁡ ( R d ) \operatorname {GL}(\mathbb {R}^d) onto the one dimensional foliations of the space of flags obtained by forgetting a single subspace.

The dimensions of these conditional measures are expressed in terms of the gap between consecutive Lyapunov exponents, and a certain entropy associated to the group action on the one dimensional foliation they are defined on. It is shown that the entropies thus defined are also related to simplicity of the Lyapunov spectrum for the given measure on GL ⁡ ( R d ) \operatorname {GL}(\mathbb {R}^d) .